If by "don't print" you mean that the thing goes through the motions -- the head and paper move -- but no ink comes out, even with a known-good cartridge, then yes there is a very common cause for this.
The flexible PCB that connects the cartridge to the logic board gets corroded at the cartridge end by ink leaking from the cartridge. Therefore no signals get to the cartridge any more.
The way to test this is to take the printer apart, unplug the flexible PCB from the logic board, and, with a good cartridge in place, to measure the resistance between the common trace (this is wider than all the others, I think it's in the middle) and each of the others. It should be quite a low resistance (10's of ohms?). If you get essentiallly infinite resistance, the flexible PCB is defective.
I have not found any way to repair it. It can be replaced, if you can get the part. HP have long since stopped supplying them. It is the same part in all Thinkjets apart from the one built into the Integral (this uses a longer flexible PCB).
If you have some other fault, let me (or the forum) know what it is, and I'll try to help.